Well I don't think voiceovers are particularly uncommon, and while they may be considered cliche in a general sense, critics and audiences are always willing to make an exception for quality.
The other boon filmmaking has is the continuity editing system where causal relationships can be set up with surprising ease. The audience has learned to pick up on mood and motivation based on these relationships. The thoughts don't need to be voiced. Examples abound, but off the top of my head, think of that famous last look in Bonnie and Clyde before the thompsons open up. We know what they're thinking and what it all means.
Since the rules of filmmaking were all pretty much set down in the silent era, we would be talking about subtitles or intertitles. Inner monologues mean a lot of text; a lot more reading than audiences were used to, and I can't see it catching on. You'd have to find some way to shorthand it, like the Darmok and Jilad aliens if you're a Star Trek fan. Intertitles were also a convention easily abandoned when sound was added, so I can't see it lasting.
But as per your request, let's say it does last; that audiences become used to knowing the inner thoughts of a character.
If anything this strengthens the privileged position of the protagonist and forces the audience to identify more closely with them. This is a dangerous tool. Consider that IOTL, without this extra motivation to identify with the protagonist, we had decades of unquestioned dominance by a particular strain of macho white male (straight of course) protagonist that still exists to this day in many ways at least as a dominant trend, if not an overriding force.
Increased call to identify with heroes of this type could push society's mood for change back by decades.
Of course it's also possible that the plots evolve similarly to OTL and that the gen pop gets deeper insight into differing voices earlier than OTL; imagine Mammy getting a say in Gone With the Wind. But I would bet that conventions stick and that only macho white straight males get the privilege of voiceover.
On the bright side the establishment would grow to recognize the fact that it's ok to have feelings and that they do have internal motivations for their actions. Poorly thought-out external placeholders might lose their hold on men's minds.